To obtain high-quality communities, the seed selection method is applied to detect the most appropriate node of the target community that contains the given node as the seed. In recent years, scholars have proposed a variety of local community detection algorithms based on local expansion. Lancichinetti et al. [11] developed an algorithm that takes random nodes as the seed. Although this method has a fast running time, it reduces the quality of the resulting community, and the test is unstable. Baumes et al. [12] proposed a robust algorithm, named IC (iterative scan), which takes a random edge as the seed. However, this method is time-consuming because it produces a great many duplicate communities when searching for seeds. Lee et al. [13] proposed a method named GCE (greedy clique expansion), which takes a k-clique as the seed. Whang et al. [14] proposed a new seeding strategy based on the same distance kernel and degree. Taking maximal cliques as the seed, Li et al. [15] introduced a deep and broad searching method to detect maximal cliques, where different communities are allowed to be merged into a larger subgraph according to some given rules. Li et al. [16] considered some eligible nodes as a seed community and expanded the seed community by absorbing adjacent nodes of the seed community based on the absorbing degree function. Zareie et al. [17] introduced a hierarchical community detection method, which proposed two indices, the topological location of a node and the closeness to the network graph core, to measure the influence of node and rank them based on these two indices. To solve the seed-dependent problem, Ding et al. [18] introduced a core detecting method to replace the seed with the core member of the target community. Mohammadi et al. [19] developed a new node ranking strategy based on nodes’ global potential values, such as influence and importance, to reveal the core of the community. Cheng et al. [20] proposed an algorithm that performs the TOPSIS (Technique for Order of Preference by Similarity to Ideal Solution) to rank each node and take the node with the highest score as the seed. Rezaei et al. [21] proposed a non-heuristic algorithm EML (Extended Machine Learning-based vital node identification), which makes use of the vitality of a part of a network for training a SVR model and predicts the vitality of each node based on this trained SVR.

The community expansion method takes a seed or a seed community as the initial community and expands it by absorbing the appropriate neighboring nodes iteratively. There are two common ways to determine the fitness of a node: running a greedy optimization process for a quality function [22–26] and spreading the influence of the seed throughout the network [27–32]. The quality function evaluates the quality of the community partition, and the scores generated by it can be used for partition ranking [2]. In a study on 13 quality functions based on 230 large real-world social, collaboration, and information networks, Yang et al. distinguished quality functions from (1) only internal community connectivity, (2) only connectivity between internal nodes and external networks; (3) both internal and external community connectivity and (4) modularity [33].

Influence spreading is a method that spreads the seed influence throughout the entire network. Inspired by the epidemic spreading model, Raghavan et al. [34] proposed the classical LPA (Label Propagation algorithm). LPA initializes each node in the network with a unique label and propagates the label throughout the network. The method stops when the node label does not change. Gregory et al. [35] proposed an improved LPA algorithm, named COPRA (Community Overlap PRopagation Algorithm), which allows overlapping by multiple labels on each node. Tang [36] proposed an algorithm based on speaker-listener label propagation to detect overlapping nodes.

In recent years, some other excellent expansion methods have been proposed. Tao et al. [37] proposed a method based on local similarity, which expands the community by connecting the node of a small-scale community with the nodes that have a high degree. Li et al. [38] proposed an algorithm to detect overlapping communities by diffusing the local spectral which is simulated by short random walks. Based on the idea that walkers who follow the same node sets should be assigned to the same community, Makoto et al. [39] proposed a retrained random-walk similarity algorithm for community expansion. Taking advantage of NGC (Nearest Greater Centrality) nodes, Luo et al. [40] proposed a novel community expansion method that adds the node that has the largest fuzzy relation with its NGC nodes into the local community. Bahadori et al. [41] proposed a probabilistic overlapping community detection method called PODCD which addresses dynamic communities efficiently. PODDCD scans communities by a probabilistic generative model which is constructed by a multi-objective optimization evolutionary-based method.

To enable readers to clearly understand recent achievements in the field of seed selection and community expansion, we summarize the characteristics, advantages, and disadvantages of the abovementioned methods in the following Table 1.

As mentioned in Section 2, researchers have proposed many local expansion algorithms and made great progress in terms of seed selection and community expansion. However, there are still problems in local expansion algorithms when determining the membership of suspicious nodes. Previous algorithms only consider the relationship between the community and suspicious nodes while ignoring the relationship between the suspicious node and its potential communities.

Let us take a brief figure legend of local community detection shown in Figure 1 to illustrate this problem. In Figure 1, subgraph C denotes the community under detection. Subgraph N denotes the neighboring nodes of C. The orange nodes in subgraph N denote the suspicious nodes of C. Subgraph P _{i(i∈{1−5})} denotes the potential communities of the suspicious nodes. The area N _o in the dotted line denotes the neighboring nodes outside C of suspicious node n. The area N _i in the dotted line denotes the neighboring nodes inside C of suspicious node n.

Node n in Figure 1 is a suspicious node of community C that is under detection. The neighboring nodes of n can be divided into two parts: subgraph N _i located in community C and subgraph N _o located in the rest of the network. Most proposed algorithms determine the membership of suspicious node n on the basis of the relationship between N _i and n. Some algorithms take the relationship between N _o and node n into consideration, but they regard N _o as a whole. Actually, N _o is not an integrated whole, where nodes likely belong to different communities. As shown in Figure 1, N _o is composed of P ₁ and P ₂, which impact n. Therefore, treating adjacent nodes as a whole in undetected areas will lead to a reduction in the accuracy of node membership judgment in the process of community detection.

Topological structure refers to describing the entities and their relationships in complex networks by two basic graphic elements: nodes and links. The more topological structure information we have, the more precisely and rapidly we can detect the community structure from the complex network. Therefore, the motivation of this paper is to explore the topological structure information and subdivide the neighborhood of suspicious nodes. Therefore, we introduce the abstract concept of potential community into the local community detection algorithm. Detailed topological structure information can be obtained by exploring the potential communities of suspicious nodes to determine the membership of suspicious nodes accurately. As shown in Figure 1, P ₁ and P ₂ are the potential communities of node n. In our method, we first calculate the similarity between P ₁, P ₂, and n. Then calculate the similarity between N _i and n. Finally, we choose the most similar one from P ₁, P ₂, and n as the result community. Thus we can obtain a more accurate result on the basis of more topological structure information.

In this paper, we use a graph G = (V, E), where V is the node set and E is the link set. The graph G can be represented as an adjacency matrix A, where A _ij denotes the connection of node i and node j, if node i and node j are connected, A _ij = 1; otherwise A _ij = 0. The graph G consists of communities, where C = {C ₁, C ₂, … , C _i }(C ₁ ∪ C ₂, … , ∪ C _i ⊆ V). A community can be represented as a node set C = {v ₁, v ₂, … , v _j }(C ∈ C, v _i ∈ V). The given node v _g is the initial node for local community detection, where v _g ∈ V. The target community C _target is a community where the given node is located in the real network, where C _target ∈ C, v _g ∈ C _target . The local community detection aims to detect the most similar community to the target community based on the given node.

The definitions related to this paper are presented in this subsection.

Definition 1

(Neighboring nodes). The neighboring nodes N(v) of node v is defined as follows: N v = u | v , u ∈ E , u ∈ V , v ∈ V (1) where node v and node v _i are connected by a link. E is the link set and V is the node set of network G.

Definition 2

(Neighboring Community). The neighboring community Γ(v) of node v is defined as follows: Γ v = N v ∪ v , v ∈ V (2)

The neighboring community of a node is the union of the node and its neighboring nodes. Figure 2 displays the nodes distribution of Karate network, and Figure 3 displays a part of Karate network. For instance, all nodes in Figure 3A make up the neighboring community of node v ₁, Γ(1) = {1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 18, 20, 22, 32}. Our algorithm takes the neighboring community of the seed node as the initial community for expansion.

Definition 3

(Community Neighbors). The community neighbors N(C) of community C is defined as follows: N C = u | u ∉ C , ∃ v ∈ C , u , v ∈ E , C ∈ V (3)

The community neighbors are neighboring nodes of community members, which are not in the community. Our algorithm takes the community neighbors of the detected community as suspicious nodes.

Definition 4

(Connected component). The connected component is defined as follows:

The connected component is a set of nodes, where there is a path between each pair of nodes. Nodes in the connected component are closely connected.

Definition 5

(Potential Community). The potential community of a node is defined as follows:

The potential communities of a node are a series of connected components that consist of the node’s neighbors. That is, every pair of nodes in the adjacent node is connected to each other.

We can search for the potential communities of node v by the following steps.

Step 1: Set up a subgraph G _v that is composed of node neighbors N(v) and links between them. That is, a tight node set formed by neighbors where each pair of nodes have a path, which is called the potential community.

Note that ⋃ P ( v ) i ∈ P ( v ) = N ( v ) , and ∀P(v) _i , P(v) _j ∈ P(v), i ≠ j, P(v) _i ∩ P(v) _j = ∅, where P(v) represents the union of P(v) _i .

Figure 3 displays the process of exploring the potential communities of node v ₁ in the Karate Club Network [42]. We use all neighboring nodes of node v ₁ to form a subgraph G v 1 . The connected components determined by the proposed algorithm are {2, 3, 4, 5, 13, 14, 18, 20, 22}, {5, 6, 7, 11}, {12} and {32}.

Definition 6

(Node Similarity). The similarity between a pair of nodes is defined as follows: N S v i , v j = Γ v j ∩ Γ v i Γ v i ∪ Γ v j , v i ∈ V , v j ∈ V (4)

The Jaccard similarity coefficient [43] is a common measure used to compare the similarity between two finite sets. It is easy to compute and has linear time complexity. Therefore, we use the Jaccard similarity coefficient value of two nodes’ neighboring communities to measure the similarity between the two nodes.

Definition 7

(Node Community Similarity). The similarity between community C and node v is defined as follows: N C S v , C = | Γ v ∩ C | × ∑ i , j ∈ Γ v ∩ C A i j d v i + d v j , v ∈ V , C ∈ V (5) where |Γ(v) ∩ C| denotes the number of nodes in the intersection of the neighboring community of node v and community C and d(v) denotes the degree of node v.

We use the internal links in the intersection of the neighboring community of the node and community as the measurement of similarity between the node and community. To distinguish the situation in which there is the same number of links in the intersection, we set the degree of nodes on both sides of the link as the weight of the link. In addition, a larger scale of intersection means a higher similarity between the node and community. Therefore, we add the number of nodes in the intersection to the formula.

Definition 8

(Fittest Community). The fittest community FC to which node v belongs is defined as follows: F C v = arg max N C S v , C , C ∈ P ∪ C detected , v ∈ V (6) where P is the potential community set of node v.

The fittest community to which a node belongs is the community that has the greatest similarity to the node among the potential communities of the node and the detected community. In the process of community expansion, nodes with the fittest community being the detected community will be added to the detected community. In addition, when the detected community and the potential communities have the greatest similarity to the node at the same time, we choose the detected community as the fittest community.

The proposed algorithm named LCDPC includes three stages: seed selection, community initialization, and community expansion. In order to give readers a clear description of the proposed algorithm, we show an example on Karate Club Network in Figures 3, 4.

Figure 3 shows an example of exploring the potential community of node v ₁. As described in Definition.5, we first get the network composed of the node v ₁ and its neighboring nodes in Figure 3A. Second, we removed node v ₁ and its links which is shown in Figure 3B. Third, each circle in Figure 3C is a connected component, the potential community of node v ₁.

When node v ₁ is the seed, the initial community of seed v ₁ is shown in Figure 4A. We can get the connected components of node v ₁ is {v ₂, v ₃, v ₄, v ₈, v ₉, v ₁₃, v ₁₄, v ₁₈, v ₂₀, v ₂₂} with similarity 3,784, {v ₅, v ₆, v ₇, v ₁₁} with similarity 450, {v ₁₂} with similarity 30 and {v ₃₂} with similarity 40 from Figure 3C. Therefore, we will form the initial community with node v ₁ and the connected component that is most similar to node v ₁.

Figure 4B shows the process of identification on the membership of node v ₅. Note that, the value of node degree is in the whole network rather than in the subgraph. First, we get the neighboring nodes of v ₅, where {v ₁} is neighbor inside the community and {v ₇, v ₁₁} are neighbors outside the community. Then, we explore the potential community of v ₅, {v ₇} and {v ₁₁}. The similarity between {v ₇} and v ₅ is (3 + 4)*2 = 14 is bigger than the similarity between {v ₁₁} and v ₅ is (3 + 3)*2 = 12. So the external similarity is 14 The internal similarity between {v ₁} and v ₅ is (16 + 3)*2 = 38 is bigger than 14. Therefore, node v ₅ is assigned to the community. If we perform the process without the application of potential community, the external neighboring nodes are considered as a whole {v ₇, v ₁₁}. The similarity between {v ₇, v ₁₁} and v ₅ is ((3 + 4) + (3 + 3))*2 = 39 which is bigger than internal similarity 38. Therefore, node v ₅ is not a member of the detected community.

There are four pseudo-code of LCDPC displaying in Algorithm 1, Algorithm 2, Algorithm 3, and Algorithm 4. The flow chart of each process is shown in Figure 5. The details of each procedure are shown in the following text.

Potential community exploration. Line three initializes a community P to the empty set to store the potential community. Line four initializes a list to store the nodes that make up the potential community. Then the first node from list is pulled out and v _temp is initialized as this node (Line 6). Lines eight to nine search for node sets that are linked in the common neighbors between the neighbors of v’ and the neighbors of v _temp and store these nodes in P and list, respectively. Lines 5–10 repeat the algorithm until there are no nodes in list. Saving a potential community P obtained above to Ps (Line 11). Lines 2–12 repeat the algorithm until each node v _i neighbors v. Line 13 returns all the potential communities Ps.

Seed selection. The seed selection procedure aims to search for the most suitable node as the seed of the target community where the given node is located. In the process of seed selection, LCDPC searches the node that meets the following two conditions: first, the degree of the node is greater than that of the given node; second, the node similarity between this node and the given node is the highest among neighboring nodes of the given node. In Algorithm 2, Line one initializes a community to the empty set to store the result community after the community expansion procedure. Line two calculates the degree of each node in the graph, which is the measurement of node similarity. Line seven calculates the neighboring nodes N (v _temp ). Line 10 calculates the similarity between the seed and each node in the neighboring nodes of the seed. Lines 8–17 search the seed based on the two conditions among N (v _temp ). The seed will be replaced iteratively by the node searched by the program above until no node meets the conditions (Lines 5–18)).

Community initialization. The community initialization procedure generates an initial community based on the seed. The initial community consists of the seed and the potential community of the seed that has the highest similarity to the seed. In Algorithm 3, Line three calculates the potential community sets P(v _seed ) of the seed based on Definition.5. Line five calculates the similarity between the seed and each potential community in P(v _seed ). The seed and the potential community with the highest similarity to the seed form the initial community (Lines 4–10).

Community expansion. The community expansion procedure adds eligible suspicious nodes to the initial community to form the resulting community. The eligible suspicious nodes should satisfy the condition that the fittest community of the node is the detected community expanded from the initial community. In Algorithm 4, the initial community generated by the community initialization procedure is assigned to the temporary community (Line 1). Line two initializes a list of suspicious nodes to the empty set. Line five obtains the community neighbors N(C _temp ) of community C _temp based on Definition.3. Line one assigns N(C _temp ) to the list of suspicious nodes. For each node in the list of suspicious nodes, LCDPC gets the potential community set (Line 9) and calculates the fittest community (Line 10). If the fittest community is the detected community, the node will be added to the detected community (Lines 11–12), and Line 13 adds the neighboring nodes that are outside the detected community to the list of suspicious nodes. The program stops when there is no suspicious node meeting the condition, which means that the community does not change anymore (Line 16).

Algorithm 1

Potential communities exploration.

Input: Graph G = {V, E}, link set E, node set V, node v.

Algorithm 2

Seed selection.

Input: Graph G = {V, E}, link set E, node set V, seed node v _seed .

Algorithm 3

Community initialization.

Input: Graph G = {V, E}, link set E, node set V, seed node v _seed .

Algorithm 4

Community expansion.

Input: Graph G = {V, E}, link set E, node set V, Initial Community C _initial .

To verify the performance of the proposed algorithms, we use the following two common evaluation criteria of community detection: NMI [44] (the normalized mutual information) and the F-measure [45].

We name the proposed algorithm that performs the process of potential community exploration to be LCDPC1 and the one that does not perform the process as LCDPC2. To verify the performance of the proposed algorithm, we compared it to eight state-of-the-art local community detection algorithms: RTLCD (a Robust Two-stage Local Community Detection algorithm) [18], Clauset [50], LWP (Luo, Wang, and Promislow) [51], Chen [52], LS (Link Similarity) [53], VI (Vertex Influence) [54], LCD (Local Community Detection based on Maximum Cliques) [55] and LCDDCE (Local Community Detection method on line graph through Degree Centrality and Expansion) [56].

Ding et al. [18] proposed a robust community detection algorithm RTLCD that consists of two stages: seed selection stage and community expansion stage. RTLCD searches the core member of the community where the given node is located in the seed selection stage, which solves the seed-dependent problem. RTLCD expands from the core member to the local community based on the relationship strength between nodes and communities in the community expansion stage, which solves the seed-invalid problem.

Based on the concept of Newman’s [57] modularity, Clauset et al. [50] introduced a local community quality function ΔR, which can be expressed as the ratio of the links within the community to the links with one or more endpoints outside the community. The Clauset algorithm adds nodes that cause the maximum increment of the quality function ΔR to the community.

Based on the Clauset algorithm, LWP et al. [51] proposed an improved quality function M, which can be expressed as the edges within the community divided by the number of edges between communities. The LWP algorithm expands the community by adding nodes that cause the maximum increment of M to the community and deletes nodes that cause the maximum increment of M but are separated from the community. Different from Clauset, LWP has definite termination criteria.

Chen et al. [52] introduced a novel quality function L, which takes the connection among nodes in the community and the connection between communities into consideration. To solve the outliers problem, the Chen algorithm checks for the changes in the quality function of the community after removing nodes.

Wu et al. [53] proposed a link similarity algorithm (LS) that can be defined as the intersection of a node’s neighbors and the nodes and neighborhoods of the community. First, greedy optimization of link similarity is performed, which adds nodes with the largest similarity to the community. Second, whether the nodes on the boundary should continue to remain in the community is checked. Third, nodes that have more neighbors in the boundary than in the community are removed.

Fanrong et al. [55] took advantage of the maximum clique expansion and proposed the LCD algorithm, which takes the maximum clique as the seeds. LCD detects all the maximum cliques in the network as seeds and expands the community from these seeds according to greedy optimization until all the maximum cliques are assigned to communities.

Yao et al. [54] proposed a variable influence local community detection algorithm (VI) based on a mechanism of influence attenuation, which is capable of detecting communities with variable scale according to the demands.

Wang et al. [56] introduced a line graph model to local community detection, and proposed an algorithm based on node centrality and PageRank. First, edges are transferred into nodes based on the line graph model. Second, nodes are ranked by a novel node similarity and PageRank and seeds are determined by this ranking. Third, the community is expanded by a fitness function.

In our experiments, all the algorithms were run in six real-world networks and three groups of LFR artificial networks, and the average of the experimental results was recorded. α in LCDDCE is set to 1.5. Note that all the algorithms that ran for more than 24 h were terminated. We take each node in the network as a given node and execute the algorithm with this given node. Finally, the results of all nodes are averaged as the performance of the algorithm on this network.

Table 5 displays NMI, Recall, Precision, F-Measure and Time metrics of the proposed algorithms with other comparison algorithms on five real-world networks. Table 6 lists the performance differences between LCDPC1 and LCDPC2 in terms of NMI, Recall, Precision and F-Measure metrics. From Table 5, we find that LCDPC1 shows better average performance in detecting communities of real networks than the other comparison algorithms. This result shows that the proposed algorithm outperforms state-of-the-art local community detection algorithms in community identification.

From Table 6, we can observe that LCDPC1 has improved in terms of NMI, Recall, Precision and F-Measure metrics compared to LCDPC2. This outcome verifies the effectiveness of the potential community application in community identification. Especially for the Karate and Dolphins networks, LCDPC1 has significant improvement compared to LCDPC2 in community identification. The common characteristic of the Karate and Dolphins networks is that they all have low mean degree values. Furthermore, we find that LCDPC1 has limited improvement compared to LCDPC2 in identifying communities on the Football and Books networks. Both the Football and Books networks have high mean degree values. These findings indicate that the potential community application has a more significant improvement in the efficiency of community identification on the network with a high mean node degree than that with a low mean node degree. In addition, LCDPC1 has good improvement on the LastFM network and has limited improvement on the Amazon network compared to LCDPC2. The LastFM and Amazon networks have the same low mean node degree, but the difference is that the LastFM has a great average community size and the Amazon network is poor. This outcome verifies that the potential community application has a more significant improvement in identifying communities on the network with a greater average community size than those with a lesser average community size.